A polynomial can be the area of one rectangle, counted piece by piece.
Split a rectangle with side lengths x+p and x+q into four rectangles. Their areas are x², px, qx, and pq. The area sum equals the product of the outer side lengths. This geometric proof applies to positive lengths; distributivity proves the identity for all real x, p, and q.
The coefficients are side lengths and sums of lengths here, not arbitrary positions on a graph.
Lengths in this area model are positive. The distributive identity itself also holds for negative real values.